Calculus Examples

$lim_{x \to 0} \frac{\cos (x) \tan (x)}{2 x}$

Step 1

Apply trigonometric identities.

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Step 1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{\cos (x) \frac{\sin (x)}{\cos (x)}}{2 x}$

Step 1.2

Cancel the common factor of $\cos (x)$ .

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Step 1.2.1

Cancel the common factor.

$lim_{x \to 0} \frac{\cos (x) \frac{\sin (x)}{\cos (x)}}{2 x}$

Step 1.2.2

Rewrite the expression.

$lim_{x \to 0} \frac{\sin (x)}{2 x}$

Step 2

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 0} \frac{\sin (x)}{x}$

Step 3

Since $\cos (x) \leq \frac{\sin (x)}{x} \leq 1$ and $lim_{x \to 0} \cos (x) = lim_{x \to 0} 1 = 1$ , apply the squeeze theorem.

$\frac{1}{2} \cdot 1$

Step 4

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$